An extension of fuzzy set, called Bipolar fuzzy set with positive and negative membership values, whose membership degree range is [-1,1]. Bipolar fuzzy set have potential impacts on many fields, including artificial intelligence, computer science, decision science etc. In this paper, we introduce the algebraic sum ⨁ and algebraic product ⨀ on Bipolar fuzzy set and Bipolar fuzzy matrices and prove that the set of all Bipolar fuzzy set and Bipolar fuzzy matrices form a commutative monoids over the above operations. In addition we discuss some properties like reflexive, symmetric, associative of the operators.